We study the local dynamics of chains of coupled nonlinear systems of second-order ordinary differential equations of diffusion-difference type. The main assumption is that the number of elements of chains is large enough. This condition allows us to pass to the problem with a continuous spatial variable. Critical cases have been considered while studying the stability of the equilibrum state. It is shown that all these cases have infinite dimension. The research technique is based on the development and application of special methods for construction of normal forms. Among the main results of the paper, we include the creation of new nonlinear boundary value problems of parabolic type, whose nonlocal dynamics describes the local behavior of solutions of the original system.

我们研究扩散差分型二阶常微分方程耦合非线性系统链的局部动力学。主要假设是链的元素数量足够多。这个条件允许我们用连续空间变量来解决问题。在研究平衡态稳定性时考虑了关键情况。结果表明，所有这些情况都具有无限维数。该研究技术基于构建范式的特殊方法的开发和应用。在本文的主要成果中，我们包括创建新的抛物型非线性边值问题，其非局部动力学描述了原始系统解的局部行为。